COSET Meaning and
Definition
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A coset refers to a fundamental concept in abstract algebra, specifically within the framework of group theory. In mathematics, a group is an algebraic structure that consists of a set of elements equipped with an operation, called a group operation, which combines any two elements of the set to produce another element of the set. A coset is a subset of a group that is formed by applying the group operation to a specific element of the group, known as a representative, and all the elements of a specific subgroup.
More formally, given a group G and a subgroup H of G, a coset of H in G is a subset of G formed by multiplying each element of H with a specific element g of G. In this case, g is the representative of the coset. The group operation is consistently applied to form all possible products between elements of H and g. Consequently, a coset includes all elements of the form hg, where h is an element of H.
Cosets possess several important properties. They have the same cardinality as the subgroup that generates them, and they are either disjoint or identical. Moreover, any element of the group G belongs to a unique coset based on the specific subgroup H. Cosets are essential tools in understanding group structures, symmetries, and various mathematical proofs related to groups. They have applications in diverse fields such as cryptography, physics, and computer science.
Common Misspellings for COSET
- xoset
- voset
- foset
- doset
- ciset
- ckset
- clset
- cpset
- c0set
- c9set
- coaet
- cozet
- coxet
- codet
- coeet
- cowet
- coswt
- cosst
- cosdt
- cosrt
Etymology of COSET
The word "coset" comes from mathematics, specifically from the field of group theory. The term was first introduced by the Hungarian mathematician Dénes Kőnig in 1913.
The term "coset" is a blend of the words "co-" and "set". The prefix "co-" in this context means "joint" or "together", indicating that the coset is formed by combining elements from two different sets. The term "set" refers to a collection of distinct objects.
In group theory, a coset is a subset of a group that is obtained by multiplying each element of a particular subgroup by a fixed element from the original group. The concept of cosets helps to partition the group into distinct subsets that share certain properties.
